Question

If the focus of a parabola divided a focal chord of the parabola in segment of length $3$ and $2$ the length of the latus rectum of the parabola is ?

• A. $\frac{3}{2}$
• B. $\frac{6}{5}$
• C. $\frac{12}{5}$
• D. $\frac{24}{4}$

Answer 1

The correct option is B $\frac{6}{5}$

Let $s(a,o)$ be the focus of the given parabola

Let the end points of the focal chord be $P(a{t}^2,2at)andQ(\frac{a}{t^2},\frac{-2a}{t})$

$SP$ and $SQ$ are segments of the focal chord with length $b$ and $c$ respectively

therefore, $b=3$ and $c=2$

also, $SP=\sqrt{{(a-at)}^2+4a^2{t}^2}=a(1+t^2)SQ=\sqrt{{(a-\frac{a}{t^2})}^2+\frac{4a^2}{t^2}}=a\left(\frac{t^2+1}{t^2}\right)$

Now we have, $\frac{1}{SP}+\frac{1}{SQ}=\frac{1}{a(1+t^2)}+\frac{t^2}{a(1+t^2)}=\frac{1}{a}\Rightarrow \frac{1}{b}+\frac{1}{c}=\frac{1}{a}\Rightarrow \frac{1}{3}+\frac{1}{2}=\frac{1}{a}\Rightarrow a=\frac{6}{5}$